In the last chapter, we (painstakingly) crafted a machine capable of computing the sum and carry of column-wise addition for a binary digit. In this chapter, we’ll synthesis the last two chapters, connecting our numeric representation in wires with our Add machine, and deal with any problems in our notation as they come up.

The first thing to notice as we put our abstract ideas into practice is that our machine diagrams are not infinitely large. Unfortunately there are infinite numbers, and because we’re representing numbers as wires (which have a size), there’s no way we can possibly fit a representation of every number into our diagram. We simply don’t have space for them all.

As a practical consideration, we’ll need to decide beforehand how many wires we’re going to dedicate to the representation of numbers. Our wire-count is going to have real, visible, implications on the largest number we can represent, which as we will see, will later affect all sorts of things we might want to do with numbers.

In the same way that choosing three decimal digits allowed us to represent \(10\times 10\times 10=1000\) different numbers, if we pick three wires to describe our binary number, we’ll get \(2\times 2\times 2=8\) numbers to play with.

\(\renewcommand{bin}[1]{#1\text{b}}\) Interestingly, we can write the equation \(2\times 2\times 2=8\) as \(\bin{10}\times\bin{10}\times\bin{10}=\bin{1000}\), which looks a lot like our original reasoning in the decimal number system. I wonder why that is?

The wire count we choose to let us describe numbers is our machines is known as our bit count, and each of the wires respectively is known as a bit. You’ve probably heard this before – a few years ago there was a lot of talk about moving all of our 32-bit computers to be 64-bit computers. Intuitively, you can think of the bit-count as controlling how big the biggest number our machines can work with.

Bits

Total Numbers

Biggest Number

\(1\)

\(2^1\)

\(1\)

\(2\)

\(2^2\)

\(3\)

\(4\)

\(2^4\)

\(15\)

\(8\)

\(2^8\)

\(255\)

\(16\)

\(2^{16}\)

\(65535\)

\(32\)

\(2^{32}\)

\(4294967295\)

\(64\)

\(2^{64}\)

\(18446744073709551615\)

\(n\)

\(2^n\)

\(2^n-1\)

As you can see, every time we add a bit (wire), we double the total count of numbers we can represent. That means when we moved from 32-bit to 64-computers, we didn’t get numbers \(2\times\) as large, we got numbers \(4294967295\times\) as large!

That’s a lot of numbers. Probably more than we’ll ever need. Definitely more than we’ll need for the purposes of this book. We’re going to arbitrarily decide on 4 bits for our machines, because it’s large enough that we can get a sense of how to work with these things, but it’s not so big that the majority of our diagrams are going to just be shuffling bits around.

In most computers, the smallest thing you can work with is 8 bits, and it’s called a byte. We’ll call the collection of our 4 bits a nybble, since it’s half a byte.

You’ll either be delighted or terribly upset that I didn’t make that joke up. That’s what these things are called.

Enough jibber-jabber. Let’s get to drawing these things. Here’s how we draw a nybble:

As a matter of convention, we’ll decide that the least-significant (lowest bit-number, therefore contributes the least to the total number) wire is on the bottom.

Remember, the way we interpret what number this nybble represents is by adding together the numbers labeled on this wires if that wire is high. If the top and bottom wires were on, but the middle two were off, this nybble would represent the number \(8+1=9\).

Let’s now finish up the work we did last chapter, and make a machine that will add two nybbles together, by using the Add machine we made!

Don’t Panic! There’s a lot going on here, but it’s actually not that bad at all.

As inputs to Add4, we’re taking two nybbles. As per our convention, the bottom-most wire in each nybble is the smallest bit in it. Both of these least-significant bits are moved to the bottom-most Add machine. Because we have nothing to “carry-in” at this point, we always input a 0 to this Cin (indicated by the circle leading into it).

The S of this Add machine is output as the least-significant bit of the nybble we’re computing. The Cout is carried over to the next adder’s Cin.

See, it’s really not very complicated conceptually; it’s just that our diagrams aren’t very powerful – they require us to draw a wire for every bit in our nybble. The problem is that we want to deal with nybbles directly, but we don’t have an abstraction mechanism for wires. In the case of machines, we can just draw a big box around the internals of it, and then use a box with a name (like we did for Add, above), which effectively allows us to abstract away the details of a machine.

Similarly, it’s pretty evident that we’re going to need a strategy for describing patterns in our wiring if we’re going to make things any larger than this Add4 machine.

Multiwires

This is a multiwire.

You can think of a multiwire as several wires all drawn together as one. The } symbol is sort of like “braiding” several wires into one single strand. In doing so, we’ve lost track of the underlying details of what the wires used to look like, which is kind of what we wanted. What we don’t want, however, is for those details to be lost entirely; we want to tuck them away for most of the time, but be able to get them back when we’re interested. Like with machines, we can hide the details by drawing a box with a label on it, but if we’re interested in how it works, we can always go back and find the diagram we drew for its internals.

To keep track of what a multiwire actually looks like in reality, we’ll annotate it. For example, a nybble would look like this:

Here, the :4 annotation on the wire indicates that this multiwire is actually a stand-in for 4 wires – ie. it’s a nybble.

Great! So multiwires let us move several wires around at once; but how do we actually work with them? The easiest way is for us to just treat them as a group of wires. For example, we could draw the Add machine (which we skipped in the last chapter, due to not wanting to have to draw all of the lines) like this:

Here we’ve annotated our multiwire with (A, B, Cout), which, as you can probably guess, means we have 3 wires “inside” of our multiwire. We’ve changed the inputs on our Sum and Cout machines to be *, which we will use to indicate that the multiwire fits “just right”.

Which is really cool! We’ve just saved ourselves a lot of time and paper for drawing these things. Unfortunately, this usage of a multiwire doesn’t fit all of our desired use-cases. Consider our implementation of Add4 above – we had to do a huge amount of wire branching in order to get everything to do what we wanted.

What we also want to do is to be able to describe “split each of the wires of this multiwire into a copy of this machine.” Such capabilities allow us to implement Add4 more concisely like this:

We use a “backwards” multiwire symbol to indicate that we want to “separate” the multiwire out and make a copy of the machine on the other end for every wire inside of the multiwire. Because both the A and B inputs have same-sized multiwires fanning into them, this means we want to process the multiwires at the same time. Meaning, we will make four copies of the Add machine. To one of them, we will take the least-significant bit from both of our multiwires and connect them to the A and B inputs, respectively. Another copy of the machine will get both second-least-significant bits, and so on.

On the other side of our Add machine, we use multiwire notation on the S output to indicate we want to put all of the S results from the machines (remember, there are 4 of them hiding in this diagram, because we make a copy for every one in the multiwire fanning into it). Therefore, the result of this machine is a nybble.

The other funky thing we need to notice is that we have a wire connecting Cin to Cout. This is clearly nonsensical, so we decide by fiat that it means “take Cout from this machine, and connect it to the Cin for the next-most-significant machine”. If there is no less-significant-machine to take a Cin from, we assume a value of 0 was passed in by default.

If this is all a little confusing to you, don’t worry. We’ve jumped up a huge level of abstraction, and that takes a lot of getting-used to. There will be a lot of practice in the upcoming chapters to make sure you’ve really nailed this stuff, but the takeaway is that the diagram we just drew out describes exactly the same thing as this one we built earlier.

In the next chapter, we’ll build some new machines to take advantage of all of this multiwire stuff, and see what else we can do with it.

Exercises

Study both the original Add4 and the multiwire Add4 diagrams. Try to get a sense of what the multiwires must be doing in order for these diagrams to be the same.

Use our new multiwire diagrams to build a machine that takes two nybbles and outputs a nybble that has performed an and gate on them, wire-wise. That is to say, the most-significant bit of your output nybble should be the and of the most-significant bits in either of the nybbles.

Assuming you have an and gate which can accept a multiwire, and outputs a single wire with value 1 if all of the wires in the multiwire were 1, construct a machine which computes whether two nybbles represent the same number.